Answer
diverges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{n^{20}+21^{n}}{n^{21}+20^{n}}$$
We use the limit comparison test with $\sum \left(\dfrac{21}{20}\right) ^n $, a divergent geometric series:
\begin{align*}
\lim_{n\to \infty} \frac{a_n}{b_n}&=\lim_{n\to \infty} \frac{n^{20}+21^{n}}{n^{21}+20^{n}}\frac{20^n}{21^n}\\
&=\lim _{n \rightarrow \infty} \frac{\frac{n^{20}}{21^ n}+1}{\frac{n^{2}}{20 ^n}+1}\\
&=1
\end{align*}
Then the given series also diverges.
